ANALYTICAL MECHANICS
where u = -. Therefore r
= dt2 ~ ldd*' dt
Substituting this value of ~ and the value of ~, which maj be obtained from equation (III), in equation (I), we have
for the equation of the orbit. When the law of force is giver / (r) is known and the orbit is determined by equation (V) On the other hand if the orbit is given equation (V) determines the law of force. Thus, if F denotes the force, p = — mf (r)
+%}• en
ILLUSTRATIVE EXAMPLE.
A particle describes a circle in a central field of force. Determine the law of force if the center of the field lies on the path.
Taking the center as the origin, Fig. 129, and the diameter through the origin as the axis and referring the circle to polar coordinates we obtain
r = 2 a cos 6,
m
FIG. 129
or
u =
2 a cos 0 for the equation of the orbit. Differentiating the last equation
o:
1
1
\a cos3 6 2 a cos 6) = 8 a V - u.